# Properties

 Label 1.2e3_5.2t1.2c1 Dimension 1 Group $C_2$ Conductor $2^{3} \cdot 5$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_2$ Conductor: $40= 2^{3} \cdot 5$ Artin number field: Splitting field of $f= x^{2} + 10$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_2$ Parity: Odd Corresponding Dirichlet character: $$\displaystyle\left(\frac{-40}{\bullet}\right)$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 7 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 2 + 3\cdot 7 + 6\cdot 7^{2} + 2\cdot 7^{3} + 2\cdot 7^{4} +O\left(7^{ 5 }\right) \\ r_{ 2 } &= 5 + 3\cdot 7 + 4\cdot 7^{3} + 4\cdot 7^{4} +O\left(7^{ 5 }\right) \\ \end{aligned}

### Generators of the action on the roots $r_{ 1 }, r_{ 2 }$

 Cycle notation $(1,2)$

### Character values on conjugacy classes

 Size Order Action on $r_{ 1 }, r_{ 2 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,2)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.